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Quantum Riemannian geometry

Beggs, Edwin J.

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Quantum Riemannian geometry

Record Type:	Electronic resources : Monograph/item
Title/Author:	Quantum Riemannian geometry/ by Edwin J. Beggs, Shahn Majid.
Author:	Beggs, Edwin J.
other author:	Majid, Shahn.
Published:	Cham :Springer International Publishing : : 2020.,
Description:	xvi, 809 p. :ill., digital ;24 cm.
[NT 15003449]:	Differentials On An Algebra -- Hopf Algebras and Their Bicovariant Calculi -- Vector Bundles and Connections -- Curvature, Cohomology and Sheaves -- Quantum Principal Bundles and Framings -- Vector Fields and Differential Operators -- Quantum Complex Structures -- Quantum Riemannian Structures -- Quantum Spacetime -- Solutions -- References -- Index.
Contained By:	Springer eBooks
Subject:	Geometry, Riemannian. -
Online resource:	https://doi.org/10.1007/978-3-030-30294-8
ISBN:	9783030302948

Quantum Riemannian geometry
Beggs, Edwin J.

Quantum Riemannian geometry[electronic resource] /by Edwin J. Beggs, Shahn Majid. - Cham :Springer International Publishing :2020. - xvi, 809 p. :ill., digital ;24 cm. - Grundlehren der mathematischen wissenschaften,v.3550072-7830 ;. - Grundlehren der mathematischen wissenschaften ;v.355..

Differentials On An Algebra -- Hopf Algebras and Their Bicovariant Calculi -- Vector Bundles and Connections -- Curvature, Cohomology and Sheaves -- Quantum Principal Bundles and Framings -- Vector Fields and Differential Operators -- Quantum Complex Structures -- Quantum Riemannian Structures -- Quantum Spacetime -- Solutions -- References -- Index.

This book provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points. Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a 'bottom up' one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum 'Levi-Civita' bimodule connection, geometric Laplacians and, in some cases, Dirac operators.The book also covers elements of Connes' approach to the subject coming from cyclic cohomology and spectral triples. Other topics include various other cohomology theories, holomorphic structures and noncommutative D-modules. A unique feature of the book is its constructive approach and its wealth of examples drawn from a large body of literature in mathematical physics, now put on a firm algebraic footing. Including exercises with solutions, it can be used as a textbook for advanced courses as well as a reference for researchers.

ISBN: 9783030302948

Standard No.: 10.1007/978-3-030-30294-8doiSubjects--Topical Terms:

519718
Geometry, Riemannian.

LC Class. No.: QA649 / .B444 2020

Dewey Class. No.: 516.373

Quantum Riemannian geometry
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100 1 $a Beggs, Edwin J. $3 3447638
245 1 0 $a Quantum Riemannian geometry $h [electronic resource] / $c by Edwin J. Beggs, Shahn Majid.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2020.
300 $a xvi, 809 p. : $b ill., digital ; $c 24 cm.
490 1 $a Grundlehren der mathematischen wissenschaften, $x 0072-7830 ; $v v.355
505 0 $a Differentials On An Algebra -- Hopf Algebras and Their Bicovariant Calculi -- Vector Bundles and Connections -- Curvature, Cohomology and Sheaves -- Quantum Principal Bundles and Framings -- Vector Fields and Differential Operators -- Quantum Complex Structures -- Quantum Riemannian Structures -- Quantum Spacetime -- Solutions -- References -- Index.
520 $a This book provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points. Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a 'bottom up' one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum 'Levi-Civita' bimodule connection, geometric Laplacians and, in some cases, Dirac operators.The book also covers elements of Connes' approach to the subject coming from cyclic cohomology and spectral triples. Other topics include various other cohomology theories, holomorphic structures and noncommutative D-modules. A unique feature of the book is its constructive approach and its wealth of examples drawn from a large body of literature in mathematical physics, now put on a firm algebraic footing. Including exercises with solutions, it can be used as a textbook for advanced courses as well as a reference for researchers.
650 0 $a Geometry, Riemannian. $3 519718
650 0 $a Geometric quantization. $3 579202
650 1 4 $a Mathematical Physics. $3 1542352
650 2 4 $a Classical and Quantum Gravitation, Relativity Theory. $3 1067066
650 2 4 $a Differential Geometry. $3 891003
650 2 4 $a Associative Rings and Algebras. $3 897405
650 2 4 $a Group Theory and Generalizations. $3 893889
700 1 $a Majid, Shahn. $3 731196
710 2 $a SpringerLink (Online service) $3 836513
773 0 $t Springer eBooks
830 0 $a Grundlehren der mathematischen wissenschaften ; $v v.355. $3 3447639
856 4 0 $u https://doi.org/10.1007/978-3-030-30294-8
950 $a Mathematics and Statistics (Springer-11649)